Mathematical modeling of complex spatio‐temporal dynamics in autocatalytic reaction‐diffusion systems with anomalous diffusion

2020 ◽  
Author(s):  
Bohdan Datsko
Keyword(s):  
Mathematical Modeling ◽  
Anomalous Diffusion ◽  
Temporal Dynamics ◽  
Reaction Diffusion ◽  
Autocatalytic Reaction ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatio Temporal

scholarly journals Stochastic simulation of the spatio-temporal dynamics of reaction-diffusion systems: the case for the bicoid gradient

2010 ◽  
Vol 7 (1) ◽  
Author(s):  
Paola Lecca ◽  
Adaoha E. C. Ihekwaba ◽  
Lorenzo Dematté ◽  
Corrado Priami
Keyword(s):  
Diffusion Coefficient ◽  
Stochastic Simulation ◽  
Diffusion Coefficients ◽  
Temporal Dynamics ◽  
Concentration Field ◽  
Reaction Diffusion ◽  
Local Concentration ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatio Temporal

SummaryReaction-diffusion systems are mathematical models that describe how the concentrations of substances distributed in space change under the influence of local chemical reactions, and diffusion which causes the substances to spread out in space. The classical representation of a reaction-diffusion system is given by semi-linear parabolic partial differential equations, whose solution predicts how diffusion causes the concentration field to change with time. This change is proportional to the diffusion coefficient. If the solute moves in a homogeneous system in thermal equilibrium, the diffusion coefficients are constants that do not depend on the local concentration of solvent and solute. However, in nonhomogeneous and structured media the assumption of constant intracellular diffusion coefficient is not necessarily valid, and, consequently, the diffusion coefficient is a function of the local concentration of solvent and solutes. In this paper we propose a stochastic model of reaction-diffusion systems, in which the diffusion coefficients are function of the local concentration, viscosity and frictional forces. We then describe the software tool Redi (REaction-DIffusion simulator) which we have developed in order to implement this model into a Gillespie-like stochastic simulation algorithm. Finally, we show the ability of our model implemented in the Redi tool to reproduce the observed gradient of the bicoid protein in the Drosophila Melanogaster embryo. With Redi, we were able to simulate with an accuracy of 1% the experimental spatio-temporal dynamics of the bicoid protein, as recorded in time-lapse experiments obtained by direct measurements of transgenic bicoidenhanced green fluorescent protein.

Turing and Benjamin–Feir instability mechanisms in non-autonomous systems

2020 ◽  
Vol 476 (2238) ◽  
pp. 20200003 ◽  
Author(s):  
Robert A. Van Gorder
Keyword(s):  
Autonomous Systems ◽  
Reaction Diffusion ◽  
Reaction Rates ◽  
Time Dependent ◽  
Model Parameters ◽  
Time Varying ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Dependent Model ◽  
Spatio Temporal

The Turing and Benjamin–Feir instabilities are two of the primary instability mechanisms useful for studying the transition from homogeneous states to heterogeneous spatial or spatio-temporal states in reaction–diffusion systems. We consider the case when the underlying reaction–diffusion system is non-autonomous or has a base state which varies in time, as in this case standard approaches, which rely on temporal eigenvalues, break down. We are able to establish respective criteria for the onset of each instability using comparison principles, obtaining inequalities which involve the in general time-dependent model parameters and their time derivatives. In the autonomous limit where the base state is constant in time, our results exactly recover the respective Turing and Benjamin–Feir conditions known in the literature. Our results make the Turing and Benjamin–Feir analysis amenable for a wide collection of applications, and allow one to better understand instabilities emergent due to a variety of non-autonomous mechanisms, including time-varying diffusion coefficients, time-varying reaction rates, time-dependent transitions between reaction kinetics and base states which change in time (such as heteroclinic connections between unique steady states, or limit cycles), to name a few examples.

SPIRAL WAVES IN MEDIA WITH COMPLEX-EXCITABLE DYNAMICS

1999 ◽  
Vol 09 (11) ◽  
pp. 2243-2247 ◽  
Author(s):  
ANDREI GORYACHEV ◽  
RAYMOND KAPRAL
Keyword(s):  
Phase Space ◽  
Temporal Dynamics ◽  
Rotational Symmetry ◽  
Reaction Diffusion ◽  
Structural Features ◽  
Spiral Waves ◽  
Space Trajectories ◽  
Reaction Diffusion Systems ◽  
Oscillatory Dynamics ◽  
Diffusion Systems

The structure of spiral waves is investigated in super-excitable reaction–diffusion systems where the local dynamics exhibits multilooped phase-space trajectories. It is shown that such systems support stable spiral waves with broken rotational symmetry and complex temporal dynamics. The main structural features of such waves, synchronization defect lines, are demonstrated to be similar to those of spiral waves in systems with complex-oscillatory dynamics.

Noise-induced suppression of periodic travelling waves in oscillatory reaction–diffusion systems

2010 ◽  
Vol 466 (2119) ◽  
pp. 1903-1917 ◽  
Author(s):  
Michael Sieber ◽  
Horst Malchow ◽  
Sergei V. Petrovskii
Keyword(s):  
Travelling Waves ◽  
Natural Populations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Oscillatory Reaction ◽  
Stochastic Forcing ◽  
Spatial Direction ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatio Temporal

Ecological field data suggest that some species show periodic changes in abundance over time and in a specific spatial direction. Periodic travelling waves as solutions to reaction–diffusion equations have helped to identify possible scenarios, by which such spatio-temporal patterns may arise. In this paper, such solutions are tested for their robustness against an irregular temporal forcing, since most natural populations can be expected to be subject to erratic fluctuations imposed by the environment. It is found that small environmental noise is able to suppress periodic travelling waves in stochastic variants of oscillatory reaction–diffusion systems. Irregular spatio-temporal oscillations, however, appear to be more robust and persist under the same stochastic forcing.

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